MCQ
If $a^2+b^2+c^2-a b-b c-c a=0$, than.
- A$c + a = b$
- B$b + c = a$
- C$a + b + c$
- ✓$a = b = c$
✓
Answer
Correct option: D.
$a = b = c$
$\text { Given: } a^2+b^2+c^2-a b-b c-c a=0$
$\Rightarrow 2\left(a^2+b^2+c^2-a b-b c-c a\right)=0$
$\Rightarrow\left(2 a^2+2 b^2+2 c^2-2 a b-2 b c-2 c a\right)=0$
$\Rightarrow\left(\left\{a^2+b^2-2 a b\right\}+\left\{b^2+c^2-2 b c\right\}+\left\{c^2+a^2-2 c a\right\}\right)=0$
$\Rightarrow(a-b)^2+(b-c)^2+(c-a)^2=0$
Now, since the sum of all squares is zero.
$\Rightarrow a - b = 0 $
$\Rightarrow a = b$
$\Rightarrow b - c = 0 $
$\Rightarrow b = c$
$\Rightarrow c - a = 0 $
$\Rightarrow c = a$
$\Rightarrow a = b = c$
$\Rightarrow 2\left(a^2+b^2+c^2-a b-b c-c a\right)=0$
$\Rightarrow\left(2 a^2+2 b^2+2 c^2-2 a b-2 b c-2 c a\right)=0$
$\Rightarrow\left(\left\{a^2+b^2-2 a b\right\}+\left\{b^2+c^2-2 b c\right\}+\left\{c^2+a^2-2 c a\right\}\right)=0$
$\Rightarrow(a-b)^2+(b-c)^2+(c-a)^2=0$
Now, since the sum of all squares is zero.
$\Rightarrow a - b = 0 $
$\Rightarrow a = b$
$\Rightarrow b - c = 0 $
$\Rightarrow b = c$
$\Rightarrow c - a = 0 $
$\Rightarrow c = a$
$\Rightarrow a = b = c$
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